Abstract Methods for improving upper and lower bounds various coverings of planar sets are proposed. New numbers partition constituents presented, suggestions the generalization presented methods offered.

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it for...

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their unive...

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This paper gives an upper bound for the first eigenvalue of the universal cover of a complete, stable minimal surface in hyperbolic space, and a sharper one for least area disks.

We give a general structure theory for reconstructing group actions on sets without any assumptions on the group, the action, or the set on which the group acts. Using certain ‘local data’ D from the action we build a group G(D) of the data and a space X (D) with an action of G(D) on X (D) that arise naturally from the data D. We use these to obtain an approximation to the original group G, the...